LAUSR

In this paper, we deal with an optimal control problem governed by the convection diffusion equations with random field in its coefficients. Mathematically, we prove the necessary and sufficient optimality conditions for the optimal control problem. Computationally, we establish a scheme to approximate the optimality system through the discretization by the upwind finite volume element method for the physical space, and by the sparse grid stochastic collocation algorithm based on the Smolyak construction for the probability space, which leads to the discrete solution of uncoupled deterministic problems. Moreover, the existence and uniqueness of the discrete solution are given. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.